Spatial Modeling

  • James P. Keener
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 20)


All of the models considered in previous chapters have relied on the implicit assumption that chemical concentrations are uniform in space. This assumption is reasonable when the region of space in which the reaction takes place is confined and quite small. However, there are many situations in which chemical concentrations are not uniform in space. A well-known example in which nonuniform distributions are crucial is the propagation of an action potential along the axon of a nerve fiber (Figure 1.1). When a nerve cell “fires,” a wave of membrane depolarization is initiated at the base of the axon (where it connects to the cell body; see Figure 2.1) and propagates along the axon out to its terminus. During propagation, large spatial gradients in membrane potential and local currents are created. The interaction between these spatial gradients and voltage-sensitive ion channels in the axonal membrane drives the wave along the axon. In order to understand the propagation of a nerve impulse, we must first master the basic principles of molecular diffusion and the interactions between chemical reaction and diffusion.


Diffusion Equation Spatial Modeling Travel Wave Solution Giant Axon Nerve Axon 
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Suggestions for Further Reading

  1. Random Walks in Biology, Howard Berg. This is a lovely introductory book on diffusion processes in biology (Berg 1993).Google Scholar
  2. Mathematical Problems in the Biological Sciences, S. Rubinow. Chapter 5 gives a nice introduction to diffusion processes (Rubinow 1973).Google Scholar
  3. Diffusional mobility of golgi proteins in membranes of living cells, N.B. Cole, C.L. Smith, N. Sciaky, M. Terasaki, and M. Edidin. This paper gives an example of how diffusion coefficients are measured in a specific biological context (Cole et al. 1996).Google Scholar
  4. Complex patterns in a simple system, John Pearson. Reaction diffusion equations are used to model many interesting phenomena. A sampler of the kinds of patterns that are seen in reaction diffusion systems is given in this paper (Pearson 1993).Google Scholar
  5. The theoretical foundation of dendritic function, Idan Segev, John Rinzel, and Gordon Shepard. This book contains the collected papers of Wilfrid Rall, a pioneer in the application of cable theory and compartment modeling to neuronal dendrites (Segev et al. 1995).Google Scholar
  6. Mathematical Physiology, James Keener and James Sneyd. Several of the topics presented in this chapter are covered here in more depth (Keener and Sneyd 1998).Google Scholar

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© Springer-Verlag New York, Inc. 2002

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  • James P. Keener

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